TY - JOUR

T1 - Short Directed Cycles in Bipartite Digraphs

AU - Seymour, Paul

AU - Spirkl, Sophie

N1 - Funding Information:
Supported by ONR grant N00014-14-1-0084 and NSF grant DMS-1265563. Acknowledgement
Publisher Copyright:
© 2020, János Bolyai Mathematical Society and Springer-Verlag.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - The Caccetta-Häggkvist conjecture implies that for every integer k ≥ 1, if G is a bipartite digraph, with n vertices in each part, and every vertex has out-degree more than n/(k+1), then G has a directed cycle of length at most 2k. If true this is best possible, and we prove this for k = 1, 2, 3, 4, 6 and all k ≥ 224,539. More generally, we conjecture that for every integer k ≥ 1, and every pair of reals α,β > 0 with kα + β > 1, if G is a bipartite digraph with bipartition (A, B), where every vertex in A has out-degree at least β|B|, and every vertex in B has out-degree at least α|A|, then G has a directed cycle of length at most 2k. This implies the Caccetta-Häggkvist conjecture (set β > 0 and very small), and again is best possible for infinitely many pairs (α,β). We prove this for k = 1,2, and prove a weaker statement (that α + β > 2/(k + 1) suffices) for k = 3,4.

AB - The Caccetta-Häggkvist conjecture implies that for every integer k ≥ 1, if G is a bipartite digraph, with n vertices in each part, and every vertex has out-degree more than n/(k+1), then G has a directed cycle of length at most 2k. If true this is best possible, and we prove this for k = 1, 2, 3, 4, 6 and all k ≥ 224,539. More generally, we conjecture that for every integer k ≥ 1, and every pair of reals α,β > 0 with kα + β > 1, if G is a bipartite digraph with bipartition (A, B), where every vertex in A has out-degree at least β|B|, and every vertex in B has out-degree at least α|A|, then G has a directed cycle of length at most 2k. This implies the Caccetta-Häggkvist conjecture (set β > 0 and very small), and again is best possible for infinitely many pairs (α,β). We prove this for k = 1,2, and prove a weaker statement (that α + β > 2/(k + 1) suffices) for k = 3,4.

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U2 - 10.1007/s00493-019-4065-5

DO - 10.1007/s00493-019-4065-5

M3 - Article

AN - SCOPUS:85083964379

VL - 40

SP - 575

EP - 599

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -